The law of the iterated logarithm and Marcinkiewicz law of large numbers forB-valuedU-statistics

1996 ◽  
Vol 9 (3) ◽  
pp. 679-701 ◽  
Author(s):  
Zhonggen Su
Keyword(s):  
Law Of Large Numbers ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law

scholarly journals On the rate of convergence in the strong law of large numbers for arrays

1992 ◽  
Vol 45 (3) ◽  
pp. 479-482 ◽  
Author(s):  
Tien-Chung Hu ◽  
N.C. Weber
Keyword(s):  
Rate Of Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Second Moment ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Independent And Identically Distributed

For sequences of independent and identically distributed random variables it is well known that the existence of the second moment implies the law of the iterated logarithm. We show that the law of the iterated logarithm does not extend to arrays of independent and identically distributed random variables and we develop an analogous rate result for such arrays under finite fourth moments.

Obtaining Prescribed Rates of Convergence for the Ergodic Theorem

1983 ◽  
Vol 35 (6) ◽  
pp. 1129-1146 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Ergodic Theorem ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Stationary Sequence ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Image Position

Let {Yn, n ∊ Z} be an ergodic strictly stationary sequence of random variables with mean zero, where Z denotes the set of integers. For n ∊ N = {1, 2, …}, let Sn = Y1 + Y2 + … + Yn. The ergodic theorem, alias the strong law of large numbers, says that n–lSn → 0 as n → ∞ a.s. If the Yn's are independent and have variance one, the law of the iterated logarithm tells us that this convergence takes place at the rate in the sense that1It is our purpose here to investigate what other rates of convergence are possible for the ergodic theorem, that is to say, what sequences {bn, n ≧ 1} have the property that2for some ergodic stationary sequence {Yn, n ∊ Z}.

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